Riemann series theorem

Riemann series theorem

Comments

Post a Comment

Popular posts from this blog

SCHLÖMILCH’S THEOREM

  SCHLÖMILCH’S THEOREM   In Section 4.3, we provide a modern English-language translation of Schlomilch paper [22]. Since the basis of his argument comes from the Mean Value Theorem for Integrals, which he refers to as the “well-known” theorem, we will provide a few preliminaries in Section 4.2 by introducing and arguing the following three results: (1) Rolle’s Theorem, which will be used to prove (2) the Mean Value Theorem, then extend that to (3) the Mean Value Theorem for Integrals.   4.1      Oscar Schlomilch   ¨ Oscar Xavier Schlomilch (13 April 1823 - 7 February 1901) was a German mathematician. ¨ He studied mathematics and physics primarily in Jena, Berlin and Vienna. Most of his work was strongly influenced by Johann Peter Gustav Lejeune Dirichlet (1805-1859). In 1844, Schlomilch received his doctorate from Friedrich-Schiller-Universität. From 1851 to 1874, he was a professor at Dresden teaching Higher Mathematics and Analytic ...
Read more

FUZZY MATHEMATICS

Fuzzy mathematics  is the branch of  mathematics  including  fuzzy set theory  and  fuzzy logic  that deals with partial inclusion of elements in a set on a spectrum, as opposed to simple binary "yes" or "no" (0 or 1) inclusion. It started in 1965 after the publication of  Lotfi Asker Zadeh 's seminal work  Fuzzy sets . [1]   Linguistics  is an example of a field that utilizes fuzzy set theory. Definition [ edit ] A  fuzzy subset   A  of a  set   X  is a  function   A :  X  →  L , where  L  is the  interval  [0, 1]. This function is also called a membership function. A membership function is a generalization of an  indicator function  (also called a  characteristic function ) of a subset defined for  L  = {0, 1}. More generally, one can use any  complete lattice   L  in a definition of a fuzzy subset  A . [2] Fuzzi...
Read more

Measure of countable unions and intersections

Image
  Basic properties [ edit ] Let  �  be a measure. Monotonicity [ edit ] If  � 1  and  � 2  are measurable sets with  � 1 ⊆ � 2  then � ( � 1 ) ≤ � ( � 2 ) . Measure of countable unions and intersections [ edit ] Subadditivity [ edit ] For any  countable   sequence   � 1 , � 2 , � 3 , …  of (not necessarily disjoint) measurable sets  � �  in  Σ : � ( ⋃ � = 1 ∞ � � ) ≤ ∑ � = 1 ∞ � ( � � ) . Continuity from below [ edit ] If  � 1 , � 2 , � 3 , …  are measurable sets that are increasing (meaning that  � 1 ⊆ � 2 ⊆ � 3 ⊆ … ) then the  union  of the sets  � �  is measurable and � ( ⋃ � = 1 ∞ � � )   =   lim � → ∞ � ( � � ) = sup � ≥ 1 � ( � � ) . Continuity from above [ edit ] If  � 1 , � 2 , � 3 , …  are measurable sets that are decreasing (meaning that  � 1 ⊇ � 2 ⊇ � 3 ⊇ … ) then the  intersection  of the sets  � �  is measurable; fur...
Read more